The fundamental theorem is the Proposition I. of the “Mémoire sur les conditions de résolubilité des équations par radicaux”; viz. given an equation of which a, b, c ... are the m roots, there is always a group of permutations of the letters a, b, c ... possessed of the following properties:—
1. Every function of the roots invariable by the substitutions of the group is rationally known.
2. Reciprocally every rationally determinable function of the roots is invariable by the substitutions of the group.
Here by an invariable function is meant not only a function of which the form is invariable by the substitutions of the group, but further, one of which the value is invariable by these substitutions: for instance, if the equation be φ(x) = 0, then φ(x) is a function of the roots invariable by any substitution whatever. And in saying that a function is rationally known, it is meant that its value is expressible rationally in terms of the coefficients and of the adjoint quantities.
For instance in the case of a general equation, the group is simply the system of the 1.2.3 ... n permutations of all the roots, since, in this case, the only rationally determinable functions are the symmetric functions of the roots.
In the case of the equation xn−1 ... + x + 1 = 0, n a prime number, a, b, c ... k = r, r g, r g² ... r g n−2, where g is a prime root of n, then the group is the cyclical group abc ... k, bc ... ka, ... kab ... j, that is, in this particular case the number of the permutations of the group is equal to the order of the equation.
This notion of the group of the original equation, or of the group of the equation as varied by the adjunction of a series of radicals, seems to be the fundamental one in Galois’s theory. But the problem of solution by radicals, instead of being the sole object of the theory, appears as the first link of a long chain of questions relating to the transformation and classification of irrationals.
Returning to the question of solution by radicals, it will be readily understood that by the adjunction of a radical the group may be diminished; for instance, in the case of the general cubic, where the group is that of the six permutations, by the adjunction of the square root which enters into the solution, the group is reduced to abc, bca, cab; that is, it becomes possible to express rationally, in terms of the coefficients and of the adjoint square root, any function such as a²b + b²c + c²a which is not altered by the cyclical substitution a into b, b into c, c into a. And hence, to determine whether an equation of a given form is solvable by radicals, the course of investigation is to inquire whether, by the successive adjunction of radicals, it is possible to reduce the original group of the equation so as to make it ultimately consist of a single permutation.
The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained—in the first instance in the form (scarcely intelligible without further explanation) that every function of the roots x1, x2 ... xn, invariable by the substitutions xak + b for xk, must be rationally known; and then in the equivalent form that the resolvent equation of the order 1.2 ... (n − 2) must have a rational root. In particular, the condition in order that a quintic equation may be solvable is that Lagrange’s resolvent of the order 6 may have a rational factor, a result obtained from a direct investigation in a valuable memoir by E. Luther, Crelle, t. xxxiv. (1847).
Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a prime number greater than 11, the depression is impossible.